Motion in a Plane · Lesson 6 of 13

Motion in a Plane with Constant Acceleration

“Derive vector equations and solve two simultaneous one-dimensional motions using a shared time.”

Learning Objectives

• Derive constant-acceleration velocity and position equations in vector form. • Resolve the equations into independent perpendicular components. • Solve for a common event time and then calculate the other coordinate. • Distinguish general definitions from formulas requiring constant acceleration.

Constant acceleration means a constant vector

A constant acceleration vector keeps both magnitude and direction fixed. In fixed Cartesian axes, each acceleration component is then constant. This is stronger than saying that the acceleration magnitude stays constant: circular motion can have a constant acceleration magnitude but a rotating direction. The usual constant-acceleration vector equations apply only in the first situation.

Velocity under constant accelerationLaTeX

a = \frac{v - u}{t} ⟹ v = u + a t

Time is measured from the initial event. The average acceleration equals the instantaneous constant acceleration.

Deriving the position equation

Because each velocity component changes linearly with time, its average over the interval equals the mean of its endpoint values. Thus average velocity is (u + v)/2 as a vector. Multiply by elapsed time to obtain displacement and substitute v = u + at. This yields r − r₀ = ut + ½at². Alternatively, integrate each component of v = u + at from 0 to t; the constant and linear terms produce the same result.

Position under constant accelerationLaTeX

r = r_{0} + u t + \frac{1}{2} a t^{2}

Differentiate to check: dr/dt = u + at. Substituting t = 0 returns the initial position r₀.

Perpendicular component equationsLaTeX

x = x_{0} + u_{x} t + \frac{1}{2} a_{x} t^{2}, y = y_{0} + u_{y} t + \frac{1}{2} a_{y} t^{2}

Use the same elapsed time in both equations. The components describe simultaneous motions of one particle.

Velocity componentsLaTeX

v_{x} = u_{x} + a_{x} t, v_{y} = u_{y} + a_{y} t

The speed is √(v_x² + v_y²), not v_x + v_y.

The component motions are independent in the sense that the x-equation contains only x-data and a_x, and the y-equation contains only y-data and a_y. They are tied together by the same clock. If x reaches a specified value at t = 6 s, evaluate y and both velocity components at that same 6 s, rather than finding an unrelated time from the y-equation.

Find time from the coordinate that is specified

A particle reaches x = 84 m

Problem
A particle starts at the origin with velocity 5 i-hat m/s and constant acceleration (3 i-hat + 2 j-hat) m/s². Find its y-coordinate and speed when x = 84 m.

1.The component equations are x = 5t + 1.5t² and y = t².
2.Set 1.5t² + 5t − 84 = 0. The future root is t = 6 s; the negative root belongs to the backward extension of the model.
3.At this same instant, y = 36 m. Velocity components are v_x = 5 + 3 × 6 = 23 m/s and v_y = 2 × 6 = 12 m/s.
4.Speed is √(23² + 12²) = √673 = 25.94 m/s. Direction is arctan(12/23) = 27.55° above +x.

Initial motion perpendicular to one acceleration component

Problem
A particle starts at the origin with velocity 10 j-hat m/s and acceleration (8 i-hat + 2 j-hat) m/s². Find y and speed when x = 16 m.

1.Here u_x = 0, so x = ½ × 8t² = 4t². Setting x = 16 gives t = 2 s for future motion.
2.The vertical coordinate is y = 10t + t² = 24 m at 2 s.
3.Velocity is 16 i-hat + 14 j-hat m/s. Speed is √(256 + 196) = √452 = 21.26 m/s.
4.The nonzero initial vertical velocity is essential; using y = ½a_y t² alone would omit 20 m of vertical displacement.

Why the vector equations do not need preferred axes

The arrows in r = r₀ + ut + ½at² represent physical vectors. Rotating the coordinate axes changes their components but not the vector relationship. Choose axes for convenience: along gravity and horizontal motion for projectiles, or north and east for navigation. The equations also extend to a third perpendicular coordinate with no change in reasoning.

What remains true for arbitrary motion

Relation	Validity
Average velocity = displacement / elapsed time	Definition; valid generally
Average acceleration = velocity change / elapsed time	Definition; valid generally
Instantaneous velocity = derivative of position	Valid wherever that derivative exists
v = u + at using one constant a	Requires constant acceleration over the interval
r = r₀ + ut + ½at²	Requires constant acceleration over the interval
Average velocity = (initial + final velocity)/2	Guaranteed for constant acceleration, not for arbitrary motion

For variable acceleration, use integration or separate intervals where a justified simpler model applies. Do not replace a changing acceleration by its final value in the constant-acceleration position formula. Also do not replace vector acceleration by its magnitude: direction is part of the information needed to describe motion in a plane.

Quick check

Two coordinate equations describe one particle. What must they share when locating the same event?

Key Takeaways

• A constant acceleration vector has fixed magnitude and direction. • The vector equations follow from the one-dimensional derivations component by component. • Use one clock to combine the coordinate motions. • A position or velocity magnitude comes from components, not their simple scalar sum. • General derivative definitions remain valid beyond constant-acceleration motion.

Previous · Lesson 5

Position, Velocity and Acceleration in a Plane

Next · Lesson 7

Relative Motion in Two Dimensions